An inverse‐free ADI algorithm for computing Lagrangian invariant subspaces. Issue 1 (21st October 2015)

• Record Type: Journal Article
• Authors: Mehrmann, Volker; Poloni, Federico
• Is Part Of: Numerical linear algebra with applications. Volume 23:Issue 1(2016:Jan.)
• Page Start: 147
• Page End: 168
• Publication Date: 2015-10-21
• Subjects: Lagrangian subspace -- permuted Lagrangian subspace -- Lyapunov equation -- Riccati equation -- low‐rank ADI method -- inverse‐free arithmetic -- permuted‐graph basis; Algebras, Linear -- Periodicals; 512.5
• DOI: 10.1002/nla.2018
• Languages: English
• ISSNs: 1070-5325

Abstract

Summary: The numerical computation of Lagrangian invariant subspaces of large‐scale Hamiltonian matrices is discussed in the context of the solution of Lyapunov equations. A new version of the low‐rank alternating direction implicit method is introduced, which, in order to avoid numerical difficulties with solutions that are of very large norm, uses an inverse‐free representation of the subspace and avoids inverses of ill‐conditioned matrices. It is shown that this prevents large growth of the elements of the solution that may destroy a low‐rank approximation of the solution. A partial error analysis is presented, and the behavior of the method is demonstrated via several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.